Rydberg Excitations in Bose-Einstein Condensates in Quasi-One-DimensionalPotentials and Optical Lattices

M. Viteau,1 M.G. Bason,1 J. Radogostowicz,2,3 N. Malossi,3 D. Ciampini,1,2,3 O. Morsch,1 and E. Arimondo1,2,3

1INO-CNR, Largo Pontecorvo 3, 56127 Pisa, Italy2Dipartimento di Fisica ‘‘E. Fermi,’’ Universita di Pisa, Largo Pontecorvo 3, 56127 Pisa, Italy

3CNISM UdR, Dipartimento di Fisica ‘‘E. Fermi,’’ Universita di Pisa, Largo Pontecorvo 3, 56127 Pisa, Italy(Received 22 March 2011; published 2 August 2011)

We experimentally realize Rydberg excitations in Bose-Einstein condensates of rubidium atoms loaded

into quasi-one-dimensional traps and in optical lattices. Our results for condensates expanded to different

sizes in the one-dimensional trap agree well with the intuitive picture of a chain of Rydberg excitations.

We also find that the Rydberg excitations in the optical lattice do not destroy the phase coherence of the

condensate, and our results in that system agree with the picture of localized collective Rydberg

excitations including nearest-neighbor blockade.

DOI: 10.1103/PhysRevLett.107.060402 PACS numbers: 03.65.Xp, 03.75.Lm

Rapid progress in cold atom physics in recent years hasled to ambitious proposals in which the strong and control-lable long-range interactions between Rydberg atoms [1]are used in order to implement quantum computationschemes [2–6] and quantum simulators [7–9]. One centralingredient of these schemes is the dipole blockade, whichhas been observed for pairs of atoms [10,11] and disor-dered ultracold atomic clouds [12–17]. In the dipole block-ade mechanism, the excitation of an atom to a Rydbergstate is inhibited if another, already excited, atom is lessthan the so-called blockade radius rb away [1].

In the case of a large number of atoms, a simple physicalpicture is that of a collection of superatoms [18] in which asingle excitation is shared among all the atoms inside theblockade radius [14,19,20]. As pointed out in Ref. [3], thissuggests that experiments requiring single atoms excited toRydberg states can also be performed with such collectiveRydberg excitations of hundreds or thousand of atoms,making it possible to work with, e.g., Bose condensates(BECs) loaded into optical lattices (OLs) without the needfor single-atom occupation of the lattice sites.

Here we demonstrate the realization of one-dimensionalchains of collective Rydberg excitations in rubidiumBECs. In addition, we study the excitation dynamics ofup to 50 Rydberg excitations in a BEC occupying around100 sites of a one-dimensional optical lattice. Classical andquantum correlations and highly entangled collectivestates are expected to be created, as pointed out inRef. [21] for one-dimensional Rydberg gases and inRef. [22] for one-dimensional OLs. Our experimentsmake it necessary to combine techniques for obtainingBECs in dipole traps and OLs with those for creating anddetecting Rydberg atoms. While the former require a vac-uum apparatus with large optical access, the latter demandprecisely controlled electric fields. In our setup (see Fig. 1),we reconciled these requirements by adding external fieldelectrodes to the quartz cell of our cold atom apparatus and

carefully studying the operating conditions under whichthe best field control and highest detection efficiency of theRydberg atoms was achieved.In our experiments we first create BECs of up to 105 87-

Rb atoms in a crossed optical dipole trap. Quasi-one-dimensional atomic samples of length l are then createdby switching off one of the trap beams and allowing theBEC to expand inside the dipole trap for up to 500 ms.Rydberg states with principal quantum number n between55 and 80 are subsequently excited by using a two-colorcoherent excitation scheme with one laser at 420 nm(obtained by frequency doubling of a 840 nm beam;beam waist 120 �m, maximum power 20 mW) blue-detuned by 0:5–1 GHz from the 6P3=2 level of

FIG. 1 (color online). Experimental setup. The Bose-Einsteincondensates are created in the optical dipole trap (only one armof the trap is shown here) and can then be expanded in the dipoletrap by switching off one of its arms and/or being loaded into theoptical lattice. The Rydberg excitation beams are almost parallelto the horizontal arm of the dipole trap, and field ionization anddetection of the Rydberg excitations is achieved by using theelectrodes (outside the quartz cell) and the channeltron, respec-tively.

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87-Rb, and a second laser at 1010–1030 nm (beam waist150 �m, maximum power 150 mW). Both lasers arelocked via a Fabry-Perot cavity to a stabilized referencelaser at 780 nm. Finally, the Rydberg atoms are detectedwith efficiency � ¼ 35� 10% [23] by field ionizationfollowed by a two-stage acceleration towards a channeltroncharge multiplier. The electrodes for field ionization andacceleration are placed outside the glass cell in which theBEC is created (the numbers of Rydberg atoms giventhroughout this Letter are corrected for �). In previousexperiments using Rydberg states with n ¼ 30–85, wedemonstrated field control at the mVcm�1 level [24].

The highly elongated atomic clouds thus created are upto l ¼ 1 mm long, while their radial dimensions are on theorder of 1–2 �m (radial dipole trap frequencies are around100 Hz). Since the expected blockade radii for the Rydbergstates used in our experiments range from 5 to 15 �m, thissuggests that at most one excitation fits radially into theBEC and that the total number of Rydberg excitationsshould depend on l. In order to test this picture, we alignthe two excitation lasers so as to be almost parallel to thedipole trap beam in which the BEC expands. After theexpansion, Rydberg atoms are created by using pulses ofup to 1:5 �s duration (during which the expansion isfrozen and Penning and blackbody ionization [25] werefound to be negligible) and finally detected by field ion-ization. The pulse duration is chosen such that the system isin the saturated regime in which the number of Rydbergatoms levels off after an initial increase on a time scale ofhundreds of nanoseconds [14]. Figure 2(a) shows typicalresults of such an experiment for excitation to the 66D5=2

state using a 1 �s excitation pulse, in which a linearincrease of the number of Rydberg atoms with the lengthof the condensate is visible. This result agrees with the

simple intuitive picture of superatoms being stacked in aone-dimensional array. The highly correlated character ofthe Rydberg excitations thus created is further demon-strated by analyzing the counting statistics of our experi-ments, which are strongly sub-Poissonian for shortexpansion times of a few milliseconds with observedMandel Q factors [26] of Qobs � �0:3, where Qobs ¼�N=hNi � 1 with �N and hNi the standard deviationand mean, respectively. By taking into account our detec-tion efficiency �, this indicates actual Q factors close to�1 (since the observed Q factor Qobs ¼ �Qact).From the data in Fig. 2(a), we can extract the mean

distance between adjacent superatoms, assuming a close-packed filling of the one-dimensional atomic cloud. To thisend we take into account that in the saturated regime theindividual superatoms perform Rabi oscillations betweentheir ground and excited states and that hence on averageonly half of the superatoms are in the excited state. InFig. 2(b), the blockade radius measured in this way isshown for different Rydberg states, together with the theo-retically expected value [1] for a pure van der Waals inter-

action following an n11=6 scaling (due to the n11 scaling ofthe C6 coefficient and the r�6 dependence of the interac-tion on the atomic distance r) and assuming a total line-width (intrinsic plus Fourier-limited) of our excitationlasers of around 300 kHz. The measured values for rb of5–15 �m confirm our interpretation of an essentially one-dimensional chain of Rydberg excitations. For high-lyingstates with n > 75we find deviations from the theoreticallyexpected scaling with n, which may be due to small electricbackground fields. By deliberately applying a small elec-tric field during the excitation pulse, we have checked that,for large n, an electric field on the order of a few mV=cmcan lead to a dipole-dipole interaction energy contribution

(scaling as n4=3) on the order of the usual van der Waalsinteraction, resulting in an effective blockade radius that issubstantially larger than what is expected for the purevan der Waals case. These observations confirm that, inspite of the compromise made in our setup in order toaccommodate the necessary laser beams as well the elec-tric field plates and the channeltron, we are able to avoidbackground electric fields larger than around 1 mV=cm bycarefully choosing the parameters for our pulse sequences.Having demonstrated the one-dimensional character of

the Rydberg excitation in the elongated condensate, wenow add an optical lattice along the direction of the one-dimensional sample by intersecting two linearly polarizedlaser beams of wavelength �L ¼ 840 nm at an angle �,leading to a lattice with spacing dL ¼ �L=½2 sinð�=2Þ�. Theoptical access in our setup allows us to realize spacings upto � 13 �m. For the smallest spacing dL ¼ �L=2 ¼0:42 �m, the maximum depth V0 of the periodic potentialVðxÞ ¼ V0 sinð2�x=dLÞ measured in units of the recoilenergy Erec ¼ @

2�2=ð2md2LÞ (where m is the mass of theatoms) is V0=Erec � 20, whereas for the largest lattice

FIG. 2 (color online). Rydberg excitation in an expanded con-densate. (a) Number of 66D5=2-Rydberg excitations (derived

from the number of detected ions and the detector efficiency)for an excitation pulse of 1 �s duration as a function of thecondensate length. The error bars indicate the standard deviationof the mean. (b) Measured blockade radius rb as a function of theprincipal quantum number n. The dashed line is the theoreticallypredicted value assuming a total laser linewidth of 300 kHz.

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spacing V0=Erec � 5000. By varying dL between 0.42 and� 13 �m, we interpolate between the extremes dL � rb,in which the Rydberg excitation on one site interactsstrongly with other excitations many sites away, and dL *rb, where only nearest neighbors interact significantly [seeFig. 3(a)]. Since in the lattice direction the size �x of the

on-site wave function is around �x ¼ dL=½�ðV0=ErecÞ1=4�,the depth of the OL is chosen to ensure that the ratio�x=dLis less than � 0:1 and hence �x � rb.

In a preliminary experiment we test the effect of severalRydberg excitation cycles on the condensate phase coher-ence between the lattice sites for dL ¼ 0:42 �m and hencedL � rb. After creating Rydberg excitations with a 1 �spulse, we perform a time-of-flight experiment by suddenlyswitching off the dipole trap and the optical lattice andimaging the atomic distribution after 23 ms of free fall.The resulting interference pattern confirms that, even afterthe projective Rydberg measurement by field ionization,the degree of phase coherence of the ground-state conden-sate wave function is not modified by the short-rangecorrelations among the atoms created by the collective

Rydberg excitations. As shown in Fig. 3(b), more than 10excitation-detection cycles can be performed on the samecondensate without a noticeable loss of phase coherence oratom number, allowing us to enhance the statistical samplesize in our experiments.In order to study the dynamics of Rydberg excitations in

the lattice in the regime dL � rb, we create a one-dimensional sample of well-defined length with a reason-ably uniform atom distribution, sharp boundaries, and acontrollable average atom number per lattice site by ex-panding the condensate for a given time, then ramping upthe power of the lattice beams, and ramping down thepower of the dipole trap beam to a value for which outsidethe lattice region atoms are lost from the trap. In the over-lap region of the lattice and the dipole trap (whose length isgiven roughly by the lattice beam diameter), the combinedradial trap depth is sufficient to hold the atoms againstgravity. The nL � 100 filled lattice sites (dL ¼ 2:27 �m ischosen for this experiment) each contain on averageNi ¼ 50–500 atoms. Since the dimensions of an on-sitewave function are around 2 �m radially and 0:2 �m in thelattice direction, we expect each lattice site to contain atmost one single collective Rydberg excitation in effec-tively ‘‘zero-dimensional’’ atomic clouds. In this systemwe measure the number of atoms in the 53D5=2 Rydberg

state as a function of time by varying the duration of theexcitation pulse [see Fig. 4(a)]. Our observations can beinterpreted by using a simple numerical model that con-tains a truncated Gaussian distribution Ni of atom numberson the lattice sites labeled by i leading to a distribution ofon-site collective Rabi frequencies

ffiffiffiffiffiffiffiffiffi�Ni

p�single, where the

single-particle Rabi frequency �single � 2�� 30 kHz for

our experimental parameters. Here � ¼ ½rb=dL� is thenearest (larger) integer number to rb=dL and describesthe number of sites taking part in a collective excitation.Another way of putting this is to observe that, since �Rydberg excitations of neighboring lattice sites aresuppressed by the dipole blockade, on average only 1=�of the lattice sites can contain an excitation. (The inter-actions between more distant neighbors through thevan der Waals interaction are smaller by a factor of 26

compared to nearest-neighbor interactions.) Taking intoaccount the average over the Rabi-flopping cycle of eachexcitation, when � ¼ 2 we expect to have around 1

4 nLcollective excitations present in the lattice in the saturatedregime.In order to visualize the scaling of the collective Rabi

frequency withffiffiffiffiffiffiNi

p, in Fig. 4(b) we plot the results of our

experiments as a function of the pulse duration normalizedto the collective Rabi frequency for the largestNi. By usingthat scaling, all the curves of Fig. 4(a) collapse onto asingle curve. One also sees a distinct change in the slopeof the excitation curves at a normalized pulse duration of0:5 �s, where the fraction of lattice sites containing aRydberg excitation is around 0.25. This point indicates

FIG. 3 (color online). Rydberg excitation in an optical lattice.(a) Schematic representation of the experiment. Depending onthe lattice spacing dL and the blockade radius rb, a collectiveRydberg excitation (indicated by the blue spheres) can either beconfined to a single lattice site (above) or extend over severallattices sites (below). (b) Time-of-flight profile of a condensatereleased from an optical lattice with dL ¼ 0:42 �m, V0=Erec �10, and rb � 6 �m before (blue line) and after ten cycles ofRydberg excitation to the 53D5=2 state (pulse duration 1 �s) and

subsequent detection of the Rydberg atoms by field ionization(red line). The virtually identical interference profiles show thatthe overall phase coherence of the condensate is not affected bythe collective Rydberg excitations.

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the crossover to the saturated regime in which the numberof excitations should, in principle, level out but in practicecontinues to grow. We believe this slow increase for longexcitation times to be due to dephasing effects induced by acoupling between the superatoms. That coupling can beintroduced into our model via a small dephasing rate �between the lattice sites, chosen such as to reproduce theobserved long-term increase in the number of collectiveexcitations. Finally, in the transient regime we see theremnants of a coherent Rabi oscillation, which is washedout due to the different local collective Rabi oscillationsadding up. The expected number of Rydberg excitations asa function of time PðtÞ is then given by

PðtÞ ¼ 1

�

X

i

sin2ð ffiffiffiffiffiffiffiffiffi�Ni

p�singletþ �NtottÞ; (1)

where Ntot ¼ PiNi is the total number of atoms and the

best fit is found for � � 10�5. The factor � outside thesummation accounts for multiple counting when �> 1. As

can be seen in Fig. 4(b), this model reproduces the basicfeatures observed in our experiment.We have demonstrated the controlled preparation of

Rydberg excitations in large ensembles of ultracold atomsin one-dimensional dipole traps and in optical lattices. Ourresults can straightforwardly be extended to two- andthree-dimensional lattice geometries, and the lattice spac-ings realizable in this work should allow single-site ad-dressing as well as selective control of the Rydbergexcitations. In particular, it will be important to verifywhether long-range crystalline order can be created byimplementing adiabatic transfer protocols as recently pro-posed in Refs. [27,28] and what kinds of effects on thecondensate coherence are to be expected, e.g., due to theRydberg-Rydberg interactions or due to the ions createdduring detection. Furthermore, appropriate ion detectiontechniques such as those involving microchannel platesshould allow direct observation of the distribution ofRydberg excitations in the lattice and in the one-dimensional trapping geometry.Financial support by the E.U.-STREP ‘‘NAMEQUAM,’’

the MIUR PRIN-2007 Project, and a CNISM ‘‘ProgettoInnesco 2007’’ is gratefully acknowledged. We thank I.Lesanovsky, T. Pohl, P. Pillet, D. Comparat, and G. Morigifor stimulating discussions, S. Rolston and R. Fazio for athorough reading of the manuscript, and P. Huillery forassistance.

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FIG. 4 (color online). Dynamics of Rydberg excitations in anoptical lattice. (a) Number of 53D5=2 Rydberg excitations as a

function of the pulse duration �p for different average atom

numbers per lattice site: hNii ¼ 500 (filled circles), hNii ¼ 200(open squares), and hNii ¼ 50 (filled triangles). (b) Here theexperimental results of (a) are plotted against the renormalized

pulse duration �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihNmaxi=hNii

p, with hNmaxi the largest atom

number. The vertical axis is scaled in terms of the fraction oflattice sites containing a collective excitation. For clarity, in (b)error bars have been omitted. The dashed line is the numericalsimulation for � ¼ 2 (see the text).

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